May 25, 2026
We study singular integral operators with variable Calderón–Zygmund kernels and their commutators with \(VMO\) functions in the framework of Orlicz spaces. After revisiting the classical \(L^p\) theory, we establish boundedness results in \(L^\Phi\) under standard \(\Delta_2\) and \(\nabla_2\) conditions on the Young function. The proofs rely on decomposition techniques and weak-type estimates. As an application, these results provide a functional-analytic foundation for a priori estimates and interior regularity of solutions to higher-order elliptic operators with discontinuous coefficients.
Regularity theory for elliptic partial differential equations (PDEs) and systems with discontinuous coefficients has a long and well-developed history (cf. e.g.,[1]–[9]). In the classical Lebesgue space setting, interior \(W^{2b,p}\) estimates for higher-order linear elliptic operators with bounded measurable or \(VMO\) coefficients were established through Korn’s method of freezing of the coefficients combined with Calderón–Zygmund singular integral theory and commutator estimates. This approach, originating in the seminal works of Chiarenza, Frasca, and Longo [10], [11], and subsequently extended by many authors, provides a sharp description of the interior behavior of solutions in the scale of \(L^p\)-based Sobolev spaces.
More precisely, for system of differential operators \(\mathcal{L}(x,D)\boldsymbol{u}=\boldsymbol{f},\) uniformly elliptic in the sense of Douglis–Nirenberg (cf. [5]) with coefficients belonging to \(VMO(\Omega)\), it is by now classical that strong solutions satisfy interior estimates of the form \[\|D^{2b}\boldsymbol{u}\|_{L^p(\Omega')} \le C\bigl( \|\boldsymbol{f}\|_{L^p(\Omega'')} + \|\boldsymbol{u}\|_{L^p(\Omega'')} \bigr), \qquad 1<p<\infty,\] for all subdomains \(\Omega'\Subset\Omega''\Subset\Omega\). These results rely essentially on the boundedness of Calderón–Zygmund operators and their commutators in \(L^p\) spaces, together with the smallness of the \(VMO\) modulus at small scales. Further, the regularity theory have been extended in the framework of Morrey and generalized Morrey-Sobolev spaces (see e.g. [8], [9] and the references therein).
Despite the completeness of the \(L^p\) theory, considerably less is known when the underlying growth is not of power type. In many problems arising in the calculus of variations, nonlinear PDEs, and models with nonstandard growth, the natural functional framework is provided by Orlicz and Orlicz–Sobolev spaces (see for instance [12] ant the references therein). However, existing regularity results in this direction are often restricted to scalar equations or second-order operators . In particular, interior regularity results for higher-order elliptic systems in general Orlicz spaces appear to be largely absent from the literature.
These spaces have emerged as a powerful generalization of the classical Lebesgue scale, capable of capturing variable growth conditions and modelling more intricate function behavior. Named after the polish mathematician who introduced them in 1932, these spaces were rigorously developed in [13], [14], while the monograph [15] presents further applications to differential equations and function theory. Orlicz spaces are particularly useful in problems exhibiting nonstandard growth, such as those arising in PDEs, fluid dynamics, and materials science.
The purpose of this paper is to extend the classical interior regularity theory for higher-order elliptic systems with \(VMO\) coefficients to a broad class of Orlicz–Sobolev spaces under minimal assumptions on the growth function.
More specifically, we consider linear elliptic systems of order \(2b\), \(b\geq 1\) integer, with matrix-valued coefficients satisfying a uniform ellipticity condition in the sense of Douglis–Nirenberg and belonging to \(VMO\cap L^\infty(\Omega)\). Assuming that the right-hand side \(\boldsymbol{f}\) of 20 belongs to an Orlicz space \(L^\Phi(\Omega;{\mathbb{R}}^m), m\geq1\), where the Young function \(\Phi\) satisfies natural structural conditions, we prove interior \(WL^{2b}_\Phi\) estimates for strong solutions.
Our main result shows that if \(\boldsymbol{u}\in WL^{2b}_\Phi(\Omega;{\mathbb{R}}^m), m\geq 1\) is a strong solution of the system 20 with \(\boldsymbol{f}\in L^\Phi(\Omega;{\mathbb{R}}^m)\), then for every pair of subdomains \(\Omega'\Subset\Omega''\Subset\Omega\) there exists a constant \(C>0\) such that \[\|\boldsymbol{u}\|_{WL^{2b}_\Phi(\Omega')} \le C\bigl( \|\boldsymbol{f}\|_{L^\Phi(\Omega'')} + \|\boldsymbol{u}\|_{L^\Phi(\Omega'')} \bigr).\] The constant \(C\) depends only on the dimension, the order of the system, the ellipticity constant, the \(L^\infty\) bounds of the coefficients, and their \(VMO\) modulus.
From a methodological point of view, the proof follows the classical strategy based on freezing the coefficients, representing solutions via fundamental solutions of constant-coefficient operators, and estimating the resulting singular integral operators and commutators. The main difficulty lies in adapting this approach to the Orlicz setting. In particular, we develop a nontrivial interpolation mechanism in Orlicz spaces satisfying \(\Delta_2\cap\nabla_2\) and combine it with a Campanato-type iteration scheme to control lower-order derivatives.
The novelty of the present work does not lie in the general philosophy of the argument, which is rooted in classical Calderón–Zygmund theory, but rather in the level of generality achieved. To the best of our knowledge, interior \(WL^{2b}_\Phi\) estimates for higher-order elliptic systems with \(VMO\) coefficients have not previously been established in the framework of general Orlicz spaces under minimal growth assumptions. Our results show that the classical \(VMO\) regularity theory is robust with respect to the underlying growth scale and naturally extends beyond the \(L^p\) setting.
The paper is organized as follows. In Sections 2 and 3 we recall basic facts on Orlicz and Orlicz–Sobolev spaces and the boundedness of singular integral operators. Section 4 is dedicated to the study of Calderón–Zygmund operators and commutators in Orlicz spaces. In Section 5 we prove the main interior regularity result for higher-order elliptic systems with \(VMO\) coefficients.
We use the following standard notation:
\(x = (x_1, \ldots, x_n) \in {\mathbb{R}}^n\) denotes a point and \(|x| = \left( \sum_{i=1}^n x_i^2 \right)^{1/2}\) is the Euclidean norm in \({\mathbb{R}}^n.\)
\({\mathcal{B}}_r(x) = \{ y \in {\mathbb{R}}^n : |y - x| < r \}\) denotes the open Euclidean ball in \({\mathbb{R}}^n\) centred at \(x\) with radius \(r > 0\). The quantity \(|{\mathcal{B}}_r(x)|=\omega_n r^n\) is its Lebesgue measure where \(\omega_n\) is the volume of the unit ball \({\mathcal{B}}_1(x)\subset {\mathbb{R}}^n\);
\({\mathbb{S}}^{n-1} = \{ x \in {\mathbb{R}}^n : |x| = 1 \}\) is the unit sphere in \({\mathbb{R}}^n\) and \(|{\mathbb{S}}^{n-1}| = n\omega_n.\)
\(L^p({\mathbb{R}}^n)\), for \(p\in[1,\infty)\), denotes the classical Lebesgue space while \(L^{p,\infty}(\mathbb{R}^n)\equiv L^p_{\text{weak}}({\mathbb{R}}^n)\), for \(1\leq p < \infty\) denotes the weak \(L^p({\mathbb{R}}^n)\) space equipped with the quasi-norm \[\|f\|_{L^p_{\text{weak}}(\mathbb{R}^n)} := \sup_{\lambda > 0} \lambda |\{ x \in \mathbb{R}^n : |f(x)| > \lambda \}|^{1/p}.\]
For any measurable function \(f \in L^1({\mathbb{R}}^n)\) and any measurable bounded domain \(D \subset {\mathbb{R}}^n\), we denote by \(f_D\) the average of \(f\) over \(D:\) \[f_D = \frac{1}{|D|} \int_D f(y)\,dy = \mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int} \vcenter{\textstyle-}\kern-.5\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int} \vcenter{\scriptstyle-}\kern-.5\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle-}\kern-.5\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle-}\kern-.5\wd 0}} \!\int_D f(y)\, dy.\]
Throughout the paper, for the sake of simplicity, we write \(\|\cdot\|_{L^\Phi(\Omega)}\) instead of \(\|\cdot\|_{L^\Phi(\Omega;\mathbb{R}^m)}\).
We are interested in the local regularity of solutions of higher-order linear elliptic PDEs. To this end, we introduce some functional spaces and integral operators that play a key role in the techniques used to obtain regularity estimates.
One of the most important operators in Harmonic Analysis is the Hardy-Littlewood maximal operator \({\mathcal{M}}f\), defined for any locally integrable function \(f \in L^1_{\mathrm{loc}}({\mathbb{R}}^n)\) by \[{\mathcal{M}}f(x) = \sup_{r>0} \mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int} \vcenter{\textstyle-}\kern-.5\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int} \vcenter{\scriptstyle-}\kern-.5\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle-}\kern-.5\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle-}\kern-.5\wd 0}} \!\int_{{\mathcal{B}}_r(x)} |f(z)| \, dz, \quad \text{ for a.e. } x\in{\mathbb{R}}^n,\] where the supremum is taken over all balls \({\mathcal{B}}_r(x)\) centered at \(x\).
The maximal function \({\mathcal{M}}f\) has several interesting properties (see, for instance [14], [16]–[18]). In particular, it is a majorant of the modulus of \(f\) almost everywhere \[|f(x)| \leq {\mathcal{M}}f(x), \quad \text{for a.e. } x \in {\mathbb{R}}^n.\]
Moreover, the strong \((p,p)\)-type Hardy-Littlewood inequality asserts that, for any \(f \in L^p({\mathbb{R}}^n)\) with \(p \in (1, \infty)\), the following inequality holds \[\label{strong1} \|{\mathcal{M}}f\|_{L^p({\mathbb{R}}^n)} \leq C_p \, \|f\|_{L^p({\mathbb{R}}^n)}.\tag{1}\]
In the case \(p=1\), the weak \((1,1)\)-type estimate holds: \[\label{weak1} \big|\{x \in {\mathbb{R}}^n : {\mathcal{M}}f(x) > t \}\big| \leq \frac{C(n)}{t} \int_{{\mathbb{R}}^n} |f(x)| \, dx, \qquad \forall \;t>0.\tag{2}\]
To describe the regularity of the coefficients, we introduce the function spaces of John-Nirenberg [19] and Sarason [20]. A measurable, locally integrable function \(a\) belongs to \(BMO\) (bounded mean oscillation) if the seminorm \[\label{BMO} \|a\|_\ast = \sup_{{\mathcal{B}}_r(x)} \mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int} \vcenter{\textstyle-}\kern-.5\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int} \vcenter{\scriptstyle-}\kern-.5\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle-}\kern-.5\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle-}\kern-.5\wd 0}} \!\int_{{\mathcal{B}}_r(x)} |a(z) - a_{{\mathcal{B}}_r(x)}| \, dz\tag{3}\] is finite. This defines a norm on \(BMO({\mathbb{R}}^n)\) modulo constants, under which it becomes a Banach space. In particular, if \(a \in L^\infty({\mathbb{R}}^n)\), then \(a \in BMO({\mathbb{R}}^n)\) so \(L^\infty({\mathbb{R}}^n) \subset BMO({\mathbb{R}}^n)\) and \(\|a\|_* \le 2\|a\|_{\infty}.\)
Moreover, \(a \in BMO\) belongs to \(VMO\) (vanishing mean oscillation) if \[\label{VMO} \gamma_a(R) = \sup_{{\mathcal{B}}(y,r), \, r \leq R} \mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int} \vcenter{\textstyle-}\kern-.5\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int} \vcenter{\scriptstyle-}\kern-.5\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle-}\kern-.5\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle-}\kern-.5\wd 0}} \!\int_{{\mathcal{B}}(y,r)} |a(z) - a_{{\mathcal{B}}(y,r)}| \, dz\tag{4}\] tends to zero as \(R\) tends to zero. The quantity \(\gamma_a(R)\) is called \(VMO\)-modulus of \(a.\)
The following results characterize \(BMO\) and \(VMO\) functions.
Lemma 1 (John–Nirenberg lemma, [19]). Let \(a \in BMO\) and \(p \in [1,\infty)\). Then for every ball \({\mathcal{B}}_r(x) \subset {\mathbb{R}}^n\), the following inequality holds: \[\label{JN} \left( \mathchoice {{\setbox 0=\displaystyle{\textstyle-}{\int} \vcenter{\textstyle-}\kern-.5\wd 0}} {{\setbox 0=\textstyle{\scriptstyle-}{\int} \vcenter{\scriptstyle-}\kern-.5\wd 0}} {{\setbox 0=\scriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle-}\kern-.5\wd 0}} {{\setbox 0=\scriptscriptstyle{\scriptscriptstyle-}{\int} \vcenter{\scriptscriptstyle-}\kern-.5\wd 0}} \!\int_{{\mathcal{B}}_r(x)} |a(z) - a_{{\mathcal{B}}_r(x)}|^p \, dz \right)^{\frac{1}{p}} \leq C(p) \|a\|_*.\qquad{(1)}\]
Theorem 2 ([20]). For any function \(f \in BMO\), the following statements are equivalent:
\(f \in VMO\);
\(f\) belongs to the \(BMO\)-closure of the set of bounded uniformly continuous functions;
\(\displaystyle \lim_{y \to 0} \| f(\cdot - y) - f(\cdot) \|_* = 0\).
We start by recalling the fundamental properties of Young functions, which play a central role in our analysis: these functions are convex and they are more flexible than the power functions \(\Phi(t)=t^p\) encountered in Lebesgue spaces.
A function \(\Phi : [0,\infty) \to [0,\infty)\) is said to be Young function if it is non-negative, strictly increasing, convex, and such that (cf. [13], [15], [21]) \[\label{limits} \lim_{t\to 0_+} \Phi(t)=0, \quad \lim_{t\to 0_+}\frac{\Phi(t)}{t} = 0, \qquad \lim_{t\to +\infty} \Phi(t)= +\infty, \quad \lim_{t\to +\infty }\frac{\Phi(t)}{t} = +\infty.\tag{5}\]
A Young function \(\Phi\) is said to satisfy:
the \(\Delta_2\)-condition if there exists a constant \(\mu > 1\) such that \[\label{def-delta2} \Phi(2t) \leq \mu \Phi(t), \quad \forall\, t > 0;\tag{6}\]
the \(\nabla_2\)-condition if there exists a constant \(l > 1\) such that \[\label{def-nabla2} \Phi(lt)\geq 2l\Phi(t), \quad \forall\, t > 0.\tag{7}\]
We write \(\Phi \in \Delta_2 \cap \nabla_2\) when both conditions are satisfied. In this case, the limits in 5 imply that \[\lim_{t \to 0_+} \Phi(t)=\Phi(0) = 0 , \qquad \lim_{t \to +\infty} \Phi(t) = +\infty.\] This meaning that the growth of \(\Phi\) is neither too slow nor too fast. As we shall see later, both conditions are crucial for the regularity results we aim to derive, hence we require that \(\Phi \in \Delta_2 \cap \nabla_2\).
For example, the Young function \(\Phi_1(t) = t^p,\) \(1 < p < \infty,\) satisfies the \(\Delta_2\)-condition with \(\mu \geq 2^p\) and the \(\nabla_2\)-condition for \(l \geq 2^{1/(p-1)}\). Other examples of Young functions are \[\Phi_2(t) = e^t - 1, \quad \Phi_3(t) = t \ln(1 + t), \qquad t \geq 0.\] It is easily seen that \(\Phi_2\) grows faster than any polynomial and satisfies \(\Phi_2\in \nabla_2\) but \(\Phi_2\notin \Delta_2\), while \(\Phi_3\) grows slower than any polynomial but faster than linearly, and therefore \(\Phi_3\in \Delta_2\) but \(\Phi_3\notin \nabla_2\).
In addition, the \(\Delta_2\)-condition implies easily (see for instance [13], [21], [22]) that for all \(t > 0\) and \(\lambda > 1\) there exists a constant \(\mu_\lambda > 1\) such that \[\label{lambda} \Phi(\lambda t) \leq \mu_\lambda \Phi(t).\tag{8}\]
The \(\nabla_2\)-condition implies quasi-convexity (see for instance [21], [22]), a characterization given in [21] will be particularly useful for our purposes.
A function \(\phi: [0,\infty) \to {\mathbb{R}}\) is called quasi-convex on \([0,\infty)\) if there exist a convex function \(\omega: [0,\infty) \to {\mathbb{R}}\) and a constant \(C_1 > 0\) such that \[\label{quasiconvexity} \omega(t) \leq \phi(t) \leq C_1 \omega(C_1 t), \qquad \forall \;t \geq 0.\tag{9}\] Every convex function is quasi-convex, but the converse is not necessarily true. The quasi-convexity admits several equivalent formulations, two of which are particularly relevant in the context of the Orlicz space theory.
Lemma 3. Let \(\Phi\) be a Young function. The following statements are equivalent:
\(\Phi \in \nabla_2\);
There exists \(\alpha \in (0,1)\) such that \(\Phi^\alpha\) is quasi-convex.
Lemma 4. Let \(\Phi\) be a Young function. The following statements are equivalent:
\(\Phi\) is quasi-convex;
There exists a constant \(\mathfrak{d} > 1\) such that \[\label{power3} \frac{\Phi(t_1)}{t_1} \leq \,\frac{\mathfrak{d}\,\Phi(\mathfrak{d} t_2)}{t_2},\qquad 0 < t_1 < t_2 < \infty.\qquad{(2)}\]
Below, for the sake of clarity, we present some of the most significant consequences of the \(\Delta_2\)- and \(\nabla_2\)-conditions for a Young function \(\Phi\) separately, as shown in [21], [22]. For this purpose, we recall the definition of a quasi-increasing function.
A function \(g: (0,\infty) \to {\mathbb{R}}\) is said to be quasi-increasing if there exists a constant \(C_2 > 0\) such that \[\label{quasincr} g(t_1) \leq C_2 g(t_2), \qquad \forall \;0 < t_1 < t_2 < \infty.\tag{10}\]
If \(\Phi\) is a Young function, then the function \(t \mapsto \frac{\Phi(t)}{t}\) is increasing on \((0,\infty)\), and hence quasi-increasing with constant \(C_2 = 1.\) Applying 10 to this function with \(t_1=t>0\) and \(t_2=C t>t_1\) with any \(C >1,\) we obtain that \(\Phi\) satisfies a quasi-convex growth condition of the form \[\label{CCC} C \Phi(t)\leq \Phi(C t), \qquad \forall \;t>0.\tag{11}\]
Lemma 5 ([15]). Let \(\Phi \in \Delta_2\). Then there exist constants \(P > 1\) and \(\mathfrak{b} > 1\) such that \[\label{power} \frac{\Phi(t_2)}{t_2^P} \leq \mathfrak{b} \frac{\Phi(t_1)}{t_1^P}, \qquad 0 < t_1 < t_2 < \infty.\qquad{(3)}\] The constants \(P\) and \(\mathfrak{b}\) depend only on the \(\Delta_2\)-constant of \(\Phi.\)
In other words, Lemma 5 asserts that there exists \(P > 1\) such that the function \(t \mapsto \frac{\Phi(t)}{t^P}\) is quasi-decreasing, that is, it satisfies ?? . In particular \(\frac{\Phi(t)}{t^P}\) is bounded from above for large \(t.\)
For our ultimate goal, we now establish a result analogous to Lemma 5 for Young functions satisfying the \(\nabla_2\)-condition.
Lemma 6. Let \(\Phi \in \nabla_2\). Then there exist constants \(R > 1\) and \(\mathfrak{a} > 1\) such that \[\label{cpower} \frac{\Phi(t_1)}{t_1^R} \leq \mathfrak{a} \, \frac{\Phi(\mathfrak{a} t_2)}{t_2^R},\qquad 0 < t_1 < t_2<\infty.\qquad{(4)}\]
Proof. Applying inequality ?? with \(0<\alpha<1\) to the quasi-convex function \(\Phi^\alpha\) from Lemma 3, there exists a constant \(\mathfrak{d} > 1\) such that \[\begin{align} \frac{\Phi^\alpha(t_1)}{t_1} \leq \mathfrak{d} \, \frac{\Phi^\alpha(\mathfrak{d} t_2)}{t_2} \qquad 0<t_1<t_2<\infty. \end{align}\] Raising both sides to the power \(1/\alpha,\) setting \[R=\frac{1}{\alpha}>1, \qquad \mathfrak{a}=\mathfrak{d}^{1/\alpha}>1,\] and using that \(\Phi\) is increasing on \((0,\infty)\) we obtain ?? which completes the proof. ◻
In particular, Lemma 6 asserts that there exists \(R>1\) such that the function \(t \mapsto \frac{\Phi(t)}{t^{R}}\) is quasi-increasing, in the sense of 10 , up to multiplicative dilation.
For our purpose (see Section 4), we need an alternative integral formulation, namely Hardy-type inequalities, involving both the \(\nabla_2\)- and \(\Delta_2\)-conditions.
Lemma 7. Let \(\Phi \in \Delta_2 \cap \nabla_2\). Then there exist exponents \(1 < r < p < \infty\) such that \[\begin{align} \int_0^t \frac{d\Phi(s)}{s^r} &\leq C_r \frac{\Phi(t)}{t^r}, \label{nabla2} \\[5pt] \int_t^\infty \frac{d\Phi(s)}{s^p} &\leq C_p \frac{\Phi(t)}{t^p}. \label{delta2} \end{align}\] {#eq: sublabel=eq:nabla2,eq:delta2}
Proof. The proof follows classical Hardy-type arguments under the \(\Delta_2\)- and \(\nabla_2\)-conditions.
Let \(\Phi \in \Delta_2 \cap \nabla_2\) and fix \(1<r<R.\) Since \(\Phi\in \nabla_2\), Lemma 6 implies \[\lim_{s \to 0_+} \frac{\Phi(s)}{s^r} = \lim_{s \to 0_+} \left(\frac{\Phi(s)}{s^R}\right) s^{R-r} = 0.\] Integrating by parts and using 8 and ?? , we obtain \[\begin{align} \int_0^t \frac{d\Phi(s)}{s^r} &\leq \frac{\Phi(t)}{t^r} + r \int_0^t \frac{\Phi(s)}{s^R} \cdot \frac{ds}{s^{r-R+1}} \\ &\leq \frac{\Phi(t)}{t^r} + r \, \frac{\mathfrak{a} \Phi(\mathfrak{a} t)}{t^R} \int_0^t \frac{ds}{s^{r-R+1}} \\ &\leq \frac{\Phi(t)}{t^r} + r \;\frac{\mathfrak{a}}{t^R} \;\mu_\mathfrak{a} \Phi(t) \;\int_0^t s^{R-r-1} \, ds \\ &= \left(1 + \frac{r}{R - r} \mathfrak{a}\mu_\mathfrak{a}\right) \frac{\Phi(t)}{t^r} =: C_r \frac{\Phi(t)}{t^r}. \end{align}\]
Now, let \(P < p < \infty.\) Since \(\Phi\in \Delta_2\), Lemma 5 yields \[\lim_{s \to \infty} \frac{\Phi(s)}{s^p} = \lim_{s \to \infty} \left(\frac{\Phi(s)}{s^P}\right) \frac{1}{s^{p - P}} = 0.\] Integrating by parts and using ?? , we obtain \[\begin{align} \int_t^\infty \frac{d\Phi(s)}{s^p} &= -\frac{\Phi(t)}{t^p} + p \int_t^\infty \frac{\Phi(s)}{s^P} \cdot \frac{ds}{s^{p - P + 1}} \\ &\leq \frac{\Phi(t)}{t^p} + p b \frac{\Phi(t)}{t^P} \int_t^\infty s^{P - p - 1} \, ds \\ &= \left(1 + \frac{p \mathfrak{b}}{p - P}\right) \frac{\Phi(t)}{t^p} =: C_{p} \frac{\Phi(t)}{t^p}. \end{align}\] ◻
Related integral Hardy inequalities under \(\Delta_2\)- and \(\nabla_2\)-conditions can be found in [23].
Corollary 8. If inequality ?? holds for some \(1<r<\infty\), then it also holds for any \(1<r_1<r\). Similarly, if inequality ?? holds for some \(1<p<\infty\), then it also holds for any \(p_1>p\).
Proof. Let \(1<r_1<r.\) For any \(0 < s \leq t\) we have \(s^{r - r_1} \leq t^{r - r_1}\). Therefore, \[\int_0^t \frac{d\Phi(s)}{s^{r_1}} = \int_0^t s^{r-r_1}\,\frac{d\Phi(s)}{s^{r}} \le t^{r-r_1} \int_0^t \frac{d\Phi(s)}{s^{r}} \le C_r\, t^{r-r_1}\frac{\Phi(t)}{t^{r}} = C_r\,\frac{\Phi(t)}{t^{r_1}},\]
Similarly, let \(p_1>p.\) For any \(t \leq s\), we have \(s^{p - p_1} \leq t^{p - p_1}\). Hence, \[\int_t^\infty \frac{d\Phi(s)}{s^{p_1}} \leq t^{p - p_1} \int_t^\infty \frac{d\Phi(s)}{s^p} \leq C_p \frac{\Phi(t)}{t^{p_1}}.\] ◻
Finally, we aim to recall the main characteristics of Orlicz spaces, and show the following theorems concerning function behavior in Orlicz spaces (see, for instance [14]). These spaces generalize the classical Lebesgue spaces, allowing for finer control over the growth behavior of functions, and are particularly well-suited for studying integral operators and PDEs with non-standard growth conditions.
The complementary Young function \(\Psi: [0,\infty) \to [0,\infty)\) associated with a Young function \(\Phi\) is defined by \[\label{Ycomplement} \Psi(y) := \sup_{x \geq 0} \left\{ x y - \Phi(x) \right\}, \qquad y \geq 0.\tag{12}\]
The function \(\Psi\) shares the main qualitative properties of \(\Phi\): it is non-negative, convex, strictly increasing, and satisfies appropriate growth conditions at zero and at infinity. For a pair of complementary Young functions \((\Phi,\Psi)\), Young’s inequality holds: \[\label{disY} xy \leq \Phi(x) + \Psi(y), \qquad x, y \in {\mathbb{R}}_+.\tag{13}\]
The Orlicz class \(\bar{L}^\Phi({\mathbb{R}}^n)\) consists of all measurable functions \(f: {\mathbb{R}}^n \to {\mathbb{R}}\) for which the Orlicz modular \[\label{modular} \rho_{\Phi}(f) := \int_{{\mathbb{R}}^n} \Phi(|f(x)|) \, dx\tag{14}\] is finite.
The Orlicz space \(L^\Phi({\mathbb{R}}^n)\) is defined as the collection of all measurable functions \(f: {\mathbb{R}}^n \to {\mathbb{R}}\) for which \(\frac{f}{\lambda} \in \bar{L}^\Phi({\mathbb{R}}^n)\) for some \(\lambda > 0.\) It is equipped with the Luxemburg norm \[\label{Lux} \|f\|_{L^\Phi({\mathbb{R}}^n)} = \inf \left\{ \lambda > 0 : \rho_\Phi\left( \frac{f}{\lambda} \right) \leq 1 \right\}.\tag{15}\]
Proposition 9 (Jensen’s Inequality in Orlicz Spaces). Let \(\Omega \subset {\mathbb{R}}^n\) be a measurable set with \(|\Omega| < \infty\), and let \(f \in L^\Phi(\Omega)\). Then \[\label{JensOrl} \Phi\left(\frac{1}{|\Omega|} \int_{\Omega} |f(x)|\, dx \right) \le \frac{1}{|\Omega|} \int_{\Omega} \Phi(|f(x)|) \, dx.\qquad{(5)}\]
Moreover, the following formulation of Jensen-type inequality in Orlicz spaces, in terms of Hardy-Littlewood maximal operator, will be useful for our results.
Proposition 10. Let \(f \in L^\Phi({\mathbb{R}}^n).\) Then, for almost every \(x\in {\mathbb{R}}^n,\) \[\label{MJens} \Phi({\mathcal{M}}f(x)) \le {\mathcal{M}}\Phi(|f|)(x),\qquad{(6)}\] where \({\mathcal{M}}\) denotes the Hardy-Littlewood maximal operator.
We recall some classical boundedness properties of the Hardy–Littlewood maximal operator on Orlicz spaces, which play a fundamental role in the study of regularity and integral estimates in nonstandard growth settings (cf. [13], [14]).
Theorem 11 (Weak-type inequality). Let \(\Phi\) be a Young function. Then there exists a constant \(C > 0\) such that, for all \(\alpha > 0\) and for all \(f \in L^\Phi({\mathbb{R}}^n)\), the Hardy–Littlewood maximal operator \({\mathcal{M}}f\) is weak \((\Phi, \Phi)\)-type, namely \[\label{weakorlicz} |\{ x \in {\mathbb{R}}^n : {\mathcal{M}}f(x) > \alpha \}| \leq \frac{C}{\Phi(\alpha)} \int_{{\mathbb{R}}^n} \Phi(|f(x)|) \, dx,\qquad{(7)}\] where \(C\) is independent of \(f\) and \(\alpha\).
Theorem 12 (Strong-type inequality). Let \(\Phi \in \nabla_2\). Then there exists a constant \(C > 0\) such that, for all \(f \in L^\Phi({\mathbb{R}}^n)\), the Hardy–Littlewood maximal operator \({\mathcal{M}}f\) is strong \((\Phi, \Phi)\)-type, i.e. the following inequality holds \[\label{strongORLnorm} \|{\mathcal{M}}f\|_{L^\Phi({\mathbb{R}}^n)} \leq C \|f\|_{L^\Phi({\mathbb{R}}^n)},\qquad{(8)}\] where \(\|\cdot\|_{L^\Phi({\mathbb{R}}^n)}\) denotes the Luxemburg norm as defined in 15 . Moreover, there exists a constant \(C'>0\) such that \[\label{strongorlicz} \int_{{\mathbb{R}}^n} \Phi({\mathcal{M}}f(x)) \, dx \leq \int_{{\mathbb{R}}^n} \Phi(C|f(x)|) \, dx.\qquad{(9)}\]
The result follows from the weak-type estimate in Theorem 11 together with the assumption \(\Phi\in\nabla_2\), which allows one to apply a standard interpolation argument in Orlicz spaces.
In this section we study the boundedness properties of singular integral operators with variable Calderón–Zygmund kernels in Orlicz spaces.
Definition 13. A function \(k(x;\xi): {\mathbb{R}}^n \times ({\mathbb{R}}^n \setminus \{0\}) \to {\mathbb{R}}\) is called a variable Calderón–Zygmund kernel (VCZ)* if the following conditions are satisfied:*
For each fixed \(x \in {\mathbb{R}}^n\), the function \(\xi \mapsto k(x;\xi)\) is a classical Calderón–Zygmund kernel, that is
\(k(x;\cdot) \in C^\infty({\mathbb{R}}^n \setminus \{0\});\)
\(k(x;\cdot)\) is homogeneous of degree \(-n\), namely \[\label{omog} k(x; \mu \xi) = \mu^{-n} k(x;\xi), \quad \forall\;\mu > 0, \; \xi \in {\mathbb{R}}^n \setminus \{0\};\qquad{(10)}\]
\(k(x;\cdot)\) satisfies the cancellation condition \[\label{delet} \int_{{\mathbb{S}}^{n-1}} k(x;\xi)\, d\sigma_\xi = 0, \quad \quad \int_{{\mathbb{S}}^{n-1}} |k(x;\xi)|\, d\sigma_\xi < \infty.\qquad{(11)}\]
For every multi–index \(\beta\), there exists a constant \(C(\beta)>0\) such that \[\label{sup} \sup_{\xi \in {\mathbb{S}}^{n-1}} \left| D^\beta_\xi k(x;\xi) \right| \leq C(\beta),\qquad{(12)}\] uniformly with respect to \(x\in {\mathbb{R}}^n\).
Lemma 14 (Hörmander condition). Let \({\mathcal{B}}={\mathcal{B}}_r(x_0)\) and \(2{\mathcal{B}}={\mathcal{B}}_{2r}(x_0).\) Then there exists a constant \(C=C(n)>0\) such that \[|k(x;x-y)-k(x_0;x_0-y)|\leq C \frac{|x_0-x|}{|x_0-y|^{n+1}}\] for all \(x\in {\mathcal{B}}\) and all \(y\notin 2{\mathcal{B}}.\)
The proof follows from the smoothness and homogeneity assumptions in Definition 13 and is standard (see, for instance, [10]).
Given a VCZ kernel \(k(x;\xi)\), we define the singular integral operators \[\begin{align} \tag{16} {\mathcal{K}}f(x) &:= \text{p.v.} \int_{{\mathbb{R}}^n} k(x; x - y) f(y)\, dy, \\ \tag{17} {\mathcal{C}}[a, f](x) &:= \text{p.v.} \int_{{\mathbb{R}}^n} k(x; x - y) [a(y) - a(x)] f(y)\, dy, \end{align}\] where \(a \in L^\infty({\mathbb{R}}^n)\) and “p.v.” denotes the Cauchy principal value.
The following theorem establishes the \(L^p\)-boundedness of the operators \({\mathcal{K}}f\) and \({\mathcal{C}}[a,f],\) and is proved in [10].
Theorem 15. Let \(1 < p < \infty\), \(f \in L^p({\mathbb{R}}^n)\), and \(a\in BMO({\mathbb{R}}^n)\). Then there exists a constant \(C = C(n, p) > 0\) e, such that \[\begin{align} \label{eq-Kf-strongp} \| {\mathcal{K}}f \|_{L^p({\mathbb{R}}^n)} &\leq C \| f \|_{L^p({\mathbb{R}}^n)}, \\ \label{eq-Caf-strongp} \| {\mathcal{C}}[a, f] \|_{L^p({\mathbb{R}}^n)} &\leq C \|a\|_*\, \| f \|_{L^p({\mathbb{R}}^n)}, \end{align}\] {#eq: sublabel=eq:eq-Kf-strongp,eq:eq-Caf-strongp} where \(\|a\|_*\) denotes the \(BMO({\mathbb{R}}^n)\) norm of \(a\).
Remark 1. The assumption \(a \in BMO({\mathbb{R}}^n)\) is sharp for the boundedness of the commutator \({\mathcal{C}}[a,f]\).
As a direct consequence of the strong \(L^p\)-boundedness, one immediately obtains the corresponding weak-type estimates (see, for instance, [17]).
Lemma 16. Let \(1 < p < \infty\) and \(f \in L^p({\mathbb{R}}^n)\). Then there exists a constant \(\kappa_p=\kappa(C,p)>0,\) where \(C\) is the constant from Theorem 15, such that \[\label{eq-Kf-weakp} \| {\mathcal{K}}f \|_{L^{p,\infty}({\mathbb{R}}^n)} \leq \kappa_p\, \| f \|_{L^p({\mathbb{R}}^n)}.\qquad{(13)}\] Equivalently, for every \(t > 0\), \[\label{eq-Kf-weakp2} \left| \left\{ x \in {\mathbb{R}}^n : |{\mathcal{K}}f(x)| > t \right\} \right| \leq \frac{\kappa_p^p}{t^p} \int_{{\mathbb{R}}^n} |f(x)|^p\, dx.\qquad{(14)}\]
Moreover, if \(a\in BMO({\mathbb{R}}^n)\), then \[\label{eq-Caf-weakp} \| {\mathcal{C}}[a, f] \|_{L^{p,\infty}({\mathbb{R}}^n)} \leq \kappa_p\, \|a\|_*\, \| f \|_{L^p({\mathbb{R}}^n)},\qquad{(15)}\] that is, for every \(t > 0\), \[\label{eq-Caf-weakp2} \left| \left\{ x \in {\mathbb{R}}^n : |{\mathcal{C}}[a, f](x)| > t \right\} \right| \leq \frac{\kappa_p^p \|a\|_*^p}{t^p} \int_{{\mathbb{R}}^n} |f(x)|^p\, dx.\qquad{(16)}\]
Proof. It is well known from the classical theory of Lorentz spaces [17] that \[L^p({\mathbb{R}}^n) \hookrightarrow L^{p,\infty}({\mathbb{R}}^n), \qquad \| g \|_{L^{p,\infty}({\mathbb{R}}^n)} \leq \| g \|_{L^p({\mathbb{R}}^n)}.\] for every \(g \in L^p({\mathbb{R}}^n).\) Applying this embedding to \(g={\mathcal{K}}f\) and \(g={\mathcal{C}}[a,f]\), and using the strong \(L^p\)-estimates in Theorem 15, yields ?? and ?? . The corresponding distributional inequalities ?? and ?? follow directly from the definition of the weak \(L^p\) norm. ◻
The \(L^p\)-boundedness of the integral operators, can be extended, under appropriate considerations, to the setting of Orlicz spaces.
Theorem 17. Let \(f \in L^\Phi({\mathbb{R}}^n)\) with \(\Phi \in \Delta_2 \cap \nabla_2.\) Assume that \(a \in BMO({\mathbb{R}}^n)\). Then the operators \({\mathcal{K}}f\) and \({\mathcal{C}}[a, f]\) are bounded on \(L^\Phi({\mathbb{R}}^n).\) More precisely, there exists a constant \(C>0\) such that \[\begin{align} \label{eq-Kfphi2} \|{\mathcal{K}}f\|_{L^\Phi({\mathbb{R}}^n)} &\leq C \|f\|_{L^\Phi({\mathbb{R}}^n)}, \\[5pt] \label{eq-CAFphi2} \|{\mathcal{C}}[a,f]\|_{L^\Phi({\mathbb{R}}^n)} &\leq C \|a\|_* \|f\|_{L^\Phi({\mathbb{R}}^n)}. \end{align}\] {#eq: sublabel=eq:eq-Kfphi2,eq:eq-CAFphi2}
Proof. Let \(f\in L^\Phi({\mathbb{R}}^n).\) We decompose \(f\) as \(f = f_t + f^t\), where \[\label{Dec} f_t(x) = \begin{cases} f(x) & \text{ if } |f(x)| \leq \dfrac{t}{C}, \\[5pt] 0 & \text{ if } |f(x)| > \dfrac{t}{C}, \end{cases}\qquad \quad f^t(x) = \begin{cases} f(x) & \text{ if } |f(x)| > \dfrac{t}{C}, \\[5pt] 0 & \text{ if } |f(x)| \leq \dfrac{t}{C}. \end{cases}\tag{18}\] Here \[C := \max\big\{ 2^{1+\frac{1}{r}} \kappa_r C_r^{\frac{1}{r}}, \, 2^{1+\frac{1}{p}} \kappa_p C_p^{\frac{1}{p}} \big\},\] where \(\kappa_r, C_r, \kappa_p, C_p\) come from Lemmas 7 and 16. By this choice, we get \[\label{C94r} \frac{1}{C94r} \leq \frac{1}{2^{r+1}\kappa_r^rC_r},\qquad \frac{1}{C^p}\leq \frac{1}{2^{p+1}\kappa_p^pC_p}.\tag{19}\]
Since \(|{\mathcal{K}}f(x)| \leq |{\mathcal{K}}f^t(x)| + |{\mathcal{K}}f_t(x)|,\) we obtain \[\int_{{\mathbb{R}}^n} \Phi(|{\mathcal{K}}f(x)|) \, dx \leq \int_{{\mathbb{R}}^n} \Phi(|{\mathcal{K}}f^t(x)|) \, dx + \int_{{\mathbb{R}}^n} \Phi(|{\mathcal{K}}f_t(x)|) \, dx =: I_1 + I_2.\]
Using the weak-type \((L^r,L^{r,\infty})\) bound for \({\mathcal{K}}\) ?? , Fubini’s theorem, and Lemma 7, we estimate \[\begin{align} I_1 &\le \int_0^\infty \bigl|\{x\in{\mathbb{R}}^n:\,|{\mathcal{K}}f^t(x)|>\tfrac{t}{2}\}\bigr|\,d\Phi(t) \\ &\le 2^r\kappa_r^r \int_0^\infty \frac{1}{t^r} \int_{\{|f(x)|>\frac{t}{C}\}} |f(x)|^r\,dx \, d\Phi(t) \\ &= 2^r\kappa_r^r \int_{{\mathbb{R}}^n}|f(x)|^r \left(\int_0^{C|f(x)|}\frac{d\Phi(t)}{t^r}\right)\,dx \\ &\le \frac{1}{2}\int_{{\mathbb{R}}^n}\Phi(C|f(x)|)\,dx. \end{align}\]
Similarly, using the weak-type \((L^p,L^{p,\infty})\) bound yields \[\begin{align} I_2 &\le 2^p\kappa_p^p \int_0^\infty \frac{1}{t^p} \int_{\{|f(x)|\le\frac{t}{C}\}} |f(x)|^p\,dx \, d\Phi(t) \\ &= 2^p\kappa_p^p \int_{{\mathbb{R}}^n}|f(x)|^p \left(\int_{C|f(x)|}^\infty\frac{d\Phi(t)}{t^p}\right)\,dx \\ &\le \frac{1}{2}\int_{{\mathbb{R}}^n}\Phi(C|f(x)|)\,dx. \end{align}\]
Combining the estimates yields \[\int_{{\mathbb{R}}^n} \Phi(|{\mathcal{K}}f(x)|) \, dx \leq \int_{{\mathbb{R}}^n} \Phi(C|f(x)|) \, dx,\] which implies \[\|{\mathcal{K}}f \|_{L^\Phi({\mathbb{R}}^n)}\le C \|f\|_{L^\Phi({\mathbb{R}}^n)}.\]
The proof for the commutator \({\mathcal{C}}[a,f]\) is analogous. In the decomposition 18 , we replace \(\frac{t}{C}\) by \(\frac{t}{C\|a\|_\ast}\) and use the weak-type bounds for \({\mathcal{C}}[a,f]\). Proceeding as above, we write \[\int_{{\mathbb{R}}^n}\Phi( |{\mathcal{C}}[a,f](x)|)\, dx\leq \int_{{\mathbb{R}}^n} \Phi( |{\mathcal{C}}[a,f^t](x)|)\, dx+ \int_{{\mathbb{R}}^n} \Phi(|\mathcal{C}[a,f_t](x)|)\, dx=:I_3+I_4.\]
Applying the weak-type estimates and the same integral technique, we obtain \[\begin{align} I_3 &\leq 2^r \kappa_r^r \|a\|^r_* \int_{{\mathbb{R}}^n} |f(x)|^r\left( \int_0^{C\|a\|_*|f(x)|} \frac{d\Phi(t)}{t^r} \right) \, dx \\ &\leq \frac{1}{2}\int_{{\mathbb{R}}^n} \Phi(C\|a\|_*|f(x)|) \,dx, \end{align}\] and \[\begin{align} I_4 &\leq 2^p \kappa_p^p \|a\|^p_* \int_{{\mathbb{R}}^n} |f(x)|^p\left( \int_{C\|a\|_*|f(x)|}^\infty \frac{d\Phi(t)}{t^p} \right) \, dx \\ &\leq \frac{1}{2}\int_{{\mathbb{R}}^n} \Phi(C\|a\|_*|f(x)|) \,dx \end{align}\]
Consequently, \[\begin{align} \int_{{\mathbb{R}}^n} \Phi(|\mathcal{C}[a,f](x)|) \,dx \leq \int_{{\mathbb{R}}^n} \Phi(C\|a\|_{*}|f(x)|) \,dx \end{align}\] which implies \[\|{\mathcal{C}}[a,f]\|_{L^\Phi({\mathbb{R}}^n)}\le C\|a\|_{*}\|f\|_{L^\Phi({\mathbb{R}}^n)}.\] This completes the proof. ◻
Let \(\Omega\subset {\mathbb{R}}^n\), with \(n \geq 3 ,\) be an open domain. Let \(\alpha = (\alpha_1, \dots, \alpha_n)\) be a multi-index of order \(|\alpha| = \alpha_1 + \dots + \alpha_n .\) We study the following linear elliptic system of order \(2b,\) with \(b\geq 1,\) \[\label{syst} {\mathcal{L}}(x, D) \boldsymbol{u} := \sum_{|\alpha| = 2b} {\mathbf{A}}_\alpha(x) \, D^\alpha \boldsymbol{u}(x) = \boldsymbol{f}(x), \qquad x\in \Omega.\tag{20}\]
Here \(\boldsymbol{u}\) is the unknown vector-valued function defined on \(\Omega\), given by \(\boldsymbol{u}=(u_1,\ldots,u_m)^\top\in{\mathbb{R}}^m\), while \(\boldsymbol{f}=(f_1,\ldots,f_m)^\top\) is a prescribed vector-valued function. Moreover, \({\mathbf{A}}_\alpha=\{a_{jk}^{(\alpha)}(x)\}_{j,k=1}^m\) denotes an \(m\times m\) matrix of measurable functions defined on \(\Omega\), where \(a^{(\alpha)}_{ij}=a^{\alpha_1\cdots\alpha_n}_{ij}\).
We use the notation \(D^\alpha\equiv D_{x_1}^{\alpha_1} \cdots D_{x_n}^{\alpha_n}\) with \(D_{x_i}=\partial/\partial x_i,\) and for brevity, we write \(D^{2b}\) to denote any derivative \(D^\alpha\) of order \(|\alpha|=2b.\)
We introduce the componentwise differential operators \[\label{eq-hom} l_{jk}(x,D)= \sum_{|\alpha|=2b}a_{jk}^{(\alpha)}(x) D^\alpha, \qquad j,k=1,\ldots m.\tag{21}\] With this notation, the system 20 can be rewritten in component forma as \[\label{syst1} \sum_{k=1}^m l_{jk}(x,D) u_k(x)=f_j(x), \qquad j=1,\ldots, m .\tag{22}\]
We assume that the coefficient functions \(a^{(\alpha)}_{jk}\in VMO\cap L^\infty(\Omega)\), with \(VMO\)-modulus \[\eta_{\mathbf{A}}(R):=\sum_{j,k=1}^\infty \sum_{|\alpha|=2b} \eta_{a^{(\alpha)}_{jk}}(R), \qquad \lim_{R\to 0} \eta_{\mathbf{A}}(R)=0,\] and we define the uniform bound \[\|{\mathbf{A}}\|_{\infty,\Omega}=\max_{j,k=1,\ldots,m}\max_{|\alpha|=2b} \|a^{(\alpha)}_{jk}\|_{\infty,\Omega}.\]
We say that \({\mathbf{u}}\in WL^{2b}_\Phi (\Omega;{\mathbb{R}}^m)\) is a local strong solution of 20 if it satisfies 22 almost everywhere in every subdomain \(\Omega'\Subset \Omega.\) Recall that the Orlicz-Sobolev quasi-norm is defined by \[\label{SobOrl} \| \boldsymbol{u} \|_{WL^{2b}_\Phi(\Omega)} := \sum_{|\alpha| \leq 2b} \| D^\alpha \boldsymbol{u} \|_{L^\Phi(\Omega)}.\tag{23}\]
We now define the principal symbol of the operator: for each \(x\in \Omega\) and \(\zeta\in {\mathbb{R}}^n,\) set \[l_{jk}(x, \zeta):=\sum_{|\alpha|=2b} a^{(\alpha)}_{jk}(x)\zeta^\alpha,\] where \(\zeta^\alpha=(\zeta_1^{\alpha_1}\cdots\zeta_n^{\alpha_n})\). We assume that the system 20 is uniformly elliptic in the Douglis-Nirenberg sense (cf. [4], [5], [8], [9]), that is, there exists a constant \(\delta>0\) such that \[\label{hom43ellip} \det \left\{ l_{jk}(x,\zeta)\right\}_{j,k=1}^m \geq \delta \, |\zeta|^{2bm} \quad \text{ for a.e.} \;x \in \Omega \text{ and all } \zeta \in {\mathbb{R}}^n.\tag{24}\]
We are now in a position to state our main regularity result.
Theorem 18. Suppose that the uniform ellipticity condition \(\eqref{hom43ellip}\) holds, and that the coefficients satisfy \(a^{(\alpha)}_{jk} \in VMO\cap L^\infty(\Omega)\), for all \(|\alpha|=2b\) and \(j,k=1,\ldots,m.\)
Assume further that \[\boldsymbol{f} \in L^\Phi(\Omega; {\mathbb{R}}^m) , \qquad \Phi \in \Delta_2 \cap \nabla_2,\] and that \(\boldsymbol{u} \in WL^{2b}_\Phi(\Omega;{\mathbb{R}}^m)\) is a strong solution of the system 20 . Then, for every pair of subdomains \(\Omega' \Subset \Omega'' \Subset \Omega,\) there exists a constant \(C>0,\) depending only on \(n,m,b,\delta, \|{\mathbf{A}}\|_{\infty,\Omega},\) and the \(VMO\)-modulus \(\eta_{\mathbf{A}},\) such that \[\label{estim1} \| \boldsymbol{u} \|_{WL^{2b}_\Phi(\Omega')} \leq C \left( \| \boldsymbol{f} \|_{L^\Phi(\Omega'')} + \| \boldsymbol{u} \|_{L^\Phi(\Omega'')} \right).\qquad{(17)}\]
Proof. To prove the theorem, we freeze the coefficients of 20 at a point \(x_0\in\Omega\) and consider the constant-coefficient elliptic differential operator of order \(2bm\) defined by \[L(x_0, D) := \det{\mathcal{L}}(x_0, D) = \det\left\{ \sum_{|\alpha|=2b} a^{(\alpha)}_{jk}(x_0) D^\alpha\right\}.\]
Since every linear differential operator with constant coefficients has a fundamental solution, we denote by \(\tilde{\Gamma}(x_0; x - y)\) a fundamental solution of \(L(x_0,D)\). Its explicit form depends on whether the dimension \(n\) is odd or even (see, for instance [5], [8], [24]). In fact, if \(n\) is odd, then \[\tilde{\Gamma}(x_0; x - y) = |x - y|^{2bm - n} P\left( x_0; \frac{x - y}{|x - y|} \right),\] where \(P(x_0; \cdot)\) is a real-analytic function on the unit sphere \({\mathbb{S}}^{n-1}\). If \(n\) is even, one introduces an auxiliary variable \(x_{n+1}\) and considers all functions \(f(x, x_{n+1})\) as being independent of \(x_{n+1}\) for fixed \(x \in {\mathbb{R}}^n\).
Let \(\{ L_{jk}(x_0, \zeta) \}_{j,k=1}^m\) denote the cofactor matrix of \(\{ l_{jk}(x_0, \zeta) \}_{j,k=1}^m.\) Observe that, for fixed \(j\) and \(k\), \(L_{jk}(x_0, D)\) is either a differential operator of order \(2b(m-1)\) or the zero operator.
Using the identity \[\sum_{k=1}^m l_{ik}(x_0, \zeta) L_{jk}(x_0, \zeta) = \delta_{ij} L(x_0, \zeta),\] we deduce that the fundamental matrix \(\Gamma(x_0; x)\) of the system 20 has entries \[\Gamma_{jk}(x_0; x) = L_{jk}(x_0, D) \tilde{\Gamma}(x_0; x).\] in which \(L_{jk}(x_0, D)\) for any fixed \(j\) and \(k\) is either a differential operator of order \(2b(m-1)\) or the zero multiplication operator. Let us note, that some cofactors may vanish identically as polynomials of \(\zeta.\) Then \(\{ L_{jk}(x_0, \zeta)\}_{j,k=1}^m\) is the cofactor matrix of \(\{ l_{jk}(x_0, \zeta)\}_{j,k=1}^m\).
Let us fix a ball \({\mathcal{B}}_r \Subset \Omega.\) It is sufficient to prove the estimate for \(\boldsymbol{v} \in C_0^\infty({\mathcal{B}}_r; {\mathbb{R}}^m),\) and then extend the result to general strong solutions by a standard density argument in Orlicz spaces with Young function satisfying \(\Delta_2\)-condition [25].
Applying \({\mathcal{L}}( x_0,D)\) to \(\boldsymbol{v},\) we obtain \[\begin{align} {\mathcal{L}}(x_0, D) \boldsymbol{v} (x) &= \left( {\mathcal{L}}(x_0, D) - {\mathcal{L}}(x, D) \right) \boldsymbol{v} (x) + {\mathcal{L}}(x, D) \boldsymbol{v}(x) \\ &= \sum_{|\alpha|=2b} \left[ {\mathbf{A}}_\alpha(x_0) - {\mathbf{A}}_\alpha(x) \right] D^\alpha \boldsymbol{v}(x) + {\mathcal{L}}(x, D) \boldsymbol{v}(x). \end{align}\]
According to the classical theory for linear elliptic operators with continuous coefficients, and its extension to operators with bounded \(VMO\) coefficients (see, e.g., [6], [8], [10]), one can represent \(\boldsymbol{v}\) locally as a Newtonian-type potential associated with the constant-coefficient operator obtained by freezing \({\mathcal{L}}(x,D)\) at \(x_0\in\Omega\), namely \[\label{eq-v} \boldsymbol{v}(x) = \int_{{\mathcal{B}}_r} \Gamma(x_0; x - y) {\mathcal{L}}(x_0, D) \boldsymbol{v}(y) \, dy.\tag{25}\]
Define \[g(y) = \begin{cases}\displaystyle \sum_{|\alpha'|=2b} \left[ {\mathbf{A}}_{\alpha'}(x_0) - {\mathbf{A}}_{\alpha'}(y) \right] D^{\alpha'} \boldsymbol{v}(y) + {\mathcal{L}}(y, D)\boldsymbol{v}(y), & y \in {\mathcal{B}}_r, \\[6pt] 0, & y \notin {\mathcal{B}}_r. \end{cases}\] Then the representation 25 can be written as \[\boldsymbol{v}(x) = (\Gamma(x_0; \cdot) * g)(x).\]
Using standard properties of the Fourier transform and integration by parts (see [10]), we obtain \[D^\alpha \boldsymbol{v}(x) = \big(D^\alpha \Gamma(x_0, \cdot) * g \big)(x) + g(x) \int_{{\mathbb{S}}^{n-1}} D^{\gamma_s} \Gamma(x_0; y) \nu_s \, d\sigma_y\] where \(\gamma_s = (\alpha_1, \dots, \alpha_{s-1}, \alpha_s - 1, \alpha_{s+1}, \dots, \alpha_n)\) is a multiindex of order \(|\gamma_s| = 2b - 1 ,\) and \(\nu_s\) is the \(s\)-th component of the outer normal to \({\mathbb{S}}^{n-1}.\)
Unfreezing the coefficients by setting \(x_0 = x\), we obtain \[\begin{align} D^\alpha \boldsymbol{v}(x) &= \text{p.v.} \int_{{\mathcal{B}}_r} D^\alpha \Gamma(x, x - y) \Bigg[ \sum_{|\alpha'|=2b} ({\mathbf{A}}_{\alpha'}(x) - {\mathbf{A}}_{\alpha'}(y)) D^{\alpha'} \boldsymbol{v}(y) \\ &\quad + \mathcal{L}(y, D)\boldsymbol{v}(y) \Bigg] dy + \mathcal{L}(x, D)\boldsymbol{v}(x) \int_{{\mathbb{S}}^{n-1}} D^{\gamma_s} \Gamma(x; y) \nu_s \, d\sigma_y. \end{align}\]
The entries \(D^{\alpha} \Gamma(x, \cdot),\) \(|\alpha|=2b\) are Calderón-Zygmund kernels (cf. [8]) and define \[\begin{align} \tag{26} {\mathcal{K}}_\alpha ({\mathcal{L}}(x,D)\boldsymbol{v})(x) &= p.v.\int_{{\mathcal{B}}_r} D^\alpha \Gamma(x, x - y) {\mathcal{L}}(y,D) \boldsymbol{v}(y) \, dy,\\ \tag{27} {\mathcal{C}}_{\alpha} [A_{\alpha'}, D^{\alpha'} \boldsymbol{v}] (x)&= p.v. \int_{{\mathcal{B}}_r} D^\alpha \Gamma(x, x - y) \left[ {\mathbf{A}}_{\alpha'}(x) - {\mathbf{A}}_{\alpha'}(y) \right] D^{\alpha'} \boldsymbol{v}(y) \, dy. \end{align}\]
Therefore, \[\begin{align} D^\alpha \boldsymbol{v}(x) = \sum_{|\alpha'|=2b} {\mathcal{C}}_{\alpha}[{\mathbf{A}}_{\alpha'}, D^{\alpha'}\boldsymbol{v}](x) + {\mathcal{K}}_\alpha ({\mathcal{L}}\boldsymbol{v}) (x) + {\mathcal{L}}\boldsymbol{v}(x) \int_{{\mathbb{S}}^{n-1}} D^{\gamma_s} \Gamma(x; y) y_s \, d\sigma_y. \end{align}\]
Using the Orlicz-space estimates ?? and ?? , together with the density of \(C_0^\infty(\Omega;{\mathbb{R}}^m)\) in \(L^\Phi(\Omega;{\mathbb{R}}^m)\) (cf. [14], [21], [26], [27]), we obtain \[\| D^{2b} \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_r)} \leq C \left( \sum_{|\alpha'|=2b} \, \| {\mathbf{A}}_{\alpha'} \|_{*}\, \| D^{\alpha'}\boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_r)} + \| {\mathcal{L}}\boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_r)} \right).\]
The \(VMO\) assumption on the higher order coefficients (cf. [7]) ensures that for each \(\varepsilon>0\) there exists \(r_0=r_0(\varepsilon, \eta_{{\mathbf{A}}_{\alpha'}})\), such that for \(r<r_0\), we have \[\| D^{2b} \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_r)} \leq C \left( \varepsilon \, \| D^{2b}\boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_r)} + \| {\mathcal{L}}\boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_r)} \right).\]
Choosing \(\varepsilon>0\) sufficiently small, we obtain \[\label{final} \| D^{2b} \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_r)} \leq C \, \| {\mathcal{L}}\boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_r)}.\tag{28}\]
Now, let \(\theta \in (0,1)\) and set \(\theta'=\frac{\theta(3-\theta)}{2} > \theta\). For a fixed ball \({\mathcal{B}}_r\Subset \Omega\) we introduce a cut-off function \(\varphi(x) \in C^\infty_0({\mathcal{B}}_r)\) such that \[\varphi(x) = \begin{cases} 1, &\;x\in {\mathcal{B}}_{\theta r}, \\ 0, &\;x\notin {\mathcal{B}}_{\theta' r}. \end{cases}\] and satisfying \[|D^s \varphi| \leq C(s) [\theta(1-\theta)r]^{-s}, \qquad 1 \leq s \leq 2b,\] since \(\theta' - \theta = \theta(1-\theta)/2\).
Since \(\boldsymbol{u} \in WL^{2b, \Phi}({\mathcal{B}}_r;{\mathbb{R}}^m)\) then also \(\boldsymbol{v} := \varphi \, \boldsymbol{u} \in WL^{2b,\Phi}({\mathcal{B}}_r;{\mathbb{R}}^m)\) and \(\boldsymbol{v}\) has a compact support in \({\mathcal{B}}_r\). Then we can apply 28 to \(\boldsymbol{v}\), obtaining \[\label{phiu} \| D^{2b} \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta r})} \leq \| D^{2b} \boldsymbol{v} \|_{L^\Phi({\mathcal{B}}_{\theta' r})} \leq C \, \| {\mathcal{L}}\boldsymbol{v} \|_{L^\Phi({\mathcal{B}}_{\theta' r})}\leq C \| {\mathcal{L}}(\varphi\boldsymbol{u}) \|_{L^\Phi({\mathcal{B}}_{\theta' r})}.\tag{29}\]
Direct calculations give \[\begin{align} {\mathcal{L}}(\varphi\boldsymbol{u})(x) & = \varphi (x)\sum_{|\alpha| = 2b} {\mathbf{A}}_\alpha(x) \, D^\alpha \boldsymbol{u}(x) + \boldsymbol{u}(x) \sum_{|\alpha| = 2b} {\mathbf{A}}_\alpha(x) \, D^\alpha \varphi(x) \\ & +\sum_{|\alpha|=2b}\sum_{0<|\beta|<2b} c_{\alpha,\beta}{\mathbf{A}}_\alpha(x)\,D^\beta\boldsymbol{u}(x)\,D^{\alpha-\beta}\varphi(x). \end{align}\]
Therefore, from inequality 29 we obtain \[\begin{align} \| D^{2b} \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta r})} \leq& C \left\{ \| \boldsymbol{f} \|_{L^\Phi({\mathcal{B}}_{\theta' r})} + \sum_{s=1}^{2b-1} \frac{\| D^s\boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta' r})}}{[\theta(1-\theta)r]^{2b-s}} + \frac{\| \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta' r})}}{[\theta(1-\theta)r]^{2b}} \right\}. \end{align}\]
Multiplying both sides by \([\theta(1-\theta)r]^{2b}\) and using \(\theta(1-\theta)\le 2\theta'(1-\theta')\), we arrive at \[\begin{align} [\theta(1-\theta)r]^{2b} &\| D^{2b} \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta r})} \leq \;C \Big\{ [\theta'(1-\theta')r]^{2b} \| \boldsymbol{f} \|_{L^\Phi({\mathcal{B}}_{\theta' r})} \\ &+ \sum_{s=1}^{2b-1} [\theta'(1-\theta')r]^s \| D^s\boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta' r})} + \| \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta' r})} \Big\}. \end{align}\]
Define the seminorms \[\Theta_s = \sup_{\theta\in(0,1)} [\theta(1-\theta)r]^s \| D^s\boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta r})},\qquad 0\leq s\leq 2b,\] then the last inequality can be rewritten as \[\label{02beta} \Theta_{2b} \le C\left\{ r^{2b}\|\boldsymbol{f}\|_{L^\Phi({\mathcal{B}}_r)} + \Theta_0 + \sum_{s=1}^{2b-1}\Theta_s \right\}.\tag{30}\]
Using the interpolation inequality in Orlicz spaces [28], we have that for every \(1\leq s\leq 2b-1\) and every \(\mu>0\) there exists a constant \(C(s)>0\) such that \[\label{interp} \|D^s\boldsymbol{u}\|_{L^\Phi({\mathcal{B}}_{\theta r})} \le \mu\,\|D^{2b}\boldsymbol{u}\|_{L^\Phi({\mathcal{B}}_{\theta r})} + \frac{C(s)}{\mu^{\frac{s}{2b-s}}}\,\|\boldsymbol{u}\|_{L^\Phi({\mathcal{B}}_{\theta r})}.\tag{31}\]
Multiplying 31 by \([\theta(1-\theta)r]^s\) and taking the supremum over \(\theta\in(0,1)\), we obtain \[\begin{align} \Theta_s &\leq \sup_{\theta \in (0,1)} \left\{ \mu [\theta(1-\theta)r]^s \| D^{2b} \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta r})} + \frac{C(s)}{\mu^{\frac{s}{2b-s}}} [\theta(1-\theta)r]^s \| \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta r})} \right\}. \end{align}\]
Choosing \(\mu := \varepsilon [\theta(1-\theta)r]^{2b-s}\) with \(\varepsilon>0\) sufficiently small, we infer for \(1\leq s \leq 2b-1\) that \[\begin{align} \Theta_{s}&\leq \sup_{\theta \in (0,1)} \Big\{ \varepsilon [\theta(1-\theta)r]^{2b} \| D^{2b} \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta r})}+ \frac{C(s) }{\varepsilon^{\frac{s}{2b-s} }} \| \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{\theta r})} \Big\} \\ &\leq \varepsilon \, \Theta_{2b} + \frac{C(s)}{\varepsilon^{\frac{s}{2b-s}}} \,\Theta_0. \end{align}\]
The inequality 30 therefore becomes \[\begin{align} \Theta_{2b} &\leq C \left\{ r^{2b} \| \boldsymbol{f} \|_{L^\Phi({\mathcal{B}}_{r})} + \Theta_0 + \sum_{s=1}^{2b-1} \left(\varepsilon \Theta_{2b} + \frac{C(s)}{\varepsilon^{\frac{s}{2b-s}}} \,\Theta_0\right) \right\} \\ &\leq C' \left\{ r^{2b} \| \boldsymbol{f} \|_{L^\Phi({\mathcal{B}}_{r})} + \Theta_0 + \varepsilon\Theta_{2b} \right\}. \end{align}\] We can absorb the last term into the left-hand side obtaining \[\Theta_{2b} \leq C \left\{r^{2b}\|\boldsymbol{f} \|_{L^\Phi({\mathcal{B}}_{r})} + \Theta_0 \right\}.\]
Recalling the definition of \(\Theta_{s}\) and fixing \(\theta=1/2\) we obtain the Cacciopoli-type estimate \[\begin{align} \label{estim2} \| D^{2b} \boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{r/2})} \leq C \left\{ \| \boldsymbol{f} \|_{L^\Phi({\mathcal{B}}_{r})} + r^{-2b} \|\boldsymbol{u} \|_{L^\Phi({\mathcal{B}}_{r})} \right\}. \end{align}\tag{32}\]
The estimate \(\eqref{estim1}\) follows from 32 by a finite covering of \(\Omega'\) with balls \({\mathcal{B}}_{r/2},\) \(r < {\rm dist}(\Omega', \partial\Omega'')\). ◻
The research of A. Gogatishvili was partially supported by the grant project 23-04720S of the Czech Science Foundation (GAČR).
The Institute of Mathematics, CAS, is supported by RVO:67985840, by Shota Rustaveli National Science Foundation (SRNSF), grant no: FR22-17770.
P. Salerno and L. Softova are members of INDAM–GNAMPA. Part of the research presented in this paper was carried out during the authors’ visit to the Institute of Mathematics of the Czech Academy of Sciences. The authors gratefully acknowledge the Institute and its staff for their hospitality and support.
The authors declare that they have no conflict of interest.
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.